On-policy distribution for Emphatic TD
I don't think the section was written in haste. I think they just didn't have space to include the whole proof. It's a bit involved, so they just gave concepts.  

An Emphatic Approach to the Problem
of Off-policy Temporal-Difference Learning gives a proof of stability. 
At least parts of it should seem familiar if you've read Sutton and Barto's proof of the convergence of linear TD(0) on page 206 of their 2nd edition RL book. 

On Convergence of Emphatic Temporal-Difference Learning gives a proof of convergence.

I confess that I don't understand these papers well enough to give a summary. If you eventually do, I would greatly appreciate an update.